Get started for free
Log In Start studying!
Get started for free Log out
Chapter 0: Problem 19
Find the slope and \(y\) -intercept. $$y-3 x=6$$
Short Answer
Expert verified
The slope is 3 and the y-intercept is 6.
Step by step solution
01
Rewrite the Equation in Slope-Intercept Form
The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept. Rewrite the given equation y - 3x = 6 in this form.
02
Solve for y
Add 3x to both sides of the equation to get y = 3x + 6. Now the equation is in slope-intercept form.
03
Identify the Slope
Compare the equation y = 3x + 6 to the general form y = mx + b. The coefficient of x is the slope, so m = 3.
04
Identify the y-intercept
Compare the equation y = 3x + 6 to the general form y = mx + b. The constant term is the y-intercept, so b = 6.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
linear equations
In mathematics, linear equations are equations of first degree. This means the highest exponent of the variable (commonly \(x\)) is 1. A linear equation generally looks like \(ax + b = 0\). In geometry, linear equations represent straight-line graphs.
For example, in the given equation \(y - 3x = 6\), it is linear because both \(y\) and \(x\) are to the first power. Converting it into \(y = 3x + 6\) doesn't change its nature; it still represents a straight line. Linear equations have many applications in different fields, including physics, economics, and biology.
slope
The slope, often denoted as \(m\), measures the steepness of a line. It is calculated as the 'rise' (the change in \(y\)) over the 'run' (the change in \(x\)). In the slope-intercept form of a linear equation, \(y = mx + b\), \(m\) is the slope.
For example, in the equation \(y = 3x + 6\), the coefficient of \(x\) is the slope. Therefore, the slope is \(m=3\). This means for every one unit increase in \(x\), \(y\) increases by 3 units. Knowing the slope helps in understanding how quickly or slowly the value of \(y\) changes with \(x\).
y-intercept
The \(y\)-intercept is where the line crosses the \(y\)-axis. In the slope-intercept form, \(y = mx + b\), \(b\) is the \(y\)-intercept.
For instance, in \(y = 3x + 6\), the constant term 6 is the \(y\)-intercept. This tells us that if \(x=0\), then \(y=6\). Therefore, the graph intersects the \(y\)-axis at the point (0, 6). The \(y\)-intercept is essential for graphing because it provides an initial point for drawing the line.
solving equations
Solving linear equations involves finding the values of variables that make the equation true. This often includes isolating the variable on one side of the equation.
For the equation \(y - 3x = 6\), adding \(3x\) to both sides rewrites it as \(y = 3x + 6\). This equation is now arranged in the slope-intercept form, making it easier to identify the slope and \(y\)-intercept.
Solving equations may involve other techniques like factoring or using the quadratic formula for different types of equations, but for linear equations, the steps are usually straightforward and involve basic algebraic operations.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
Get started for free
Most popular questions from this chapter
Recommended explanations on Math Textbooks
Probability and Statistics
Read ExplanationCalculus
Read ExplanationTheoretical and Mathematical Physics
Read ExplanationMechanics Maths
Read ExplanationApplied Mathematics
Read ExplanationLogic and Functions
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.